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A base-10 integer n can be represented as $32_a$ in one base and $13_b$ in another base, where a and b are any integer bases larger than 3. What is the smallest possible sum a+b?

 Jun 25, 2022
 #1
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We have: \(3a + 2 = b + 3\)

 

Rearrange as: \(3a = b + 1\)

 

We want a to be as small as possible, so \(a = 4\), meaning \(b = 11\)

 

This means that the sum is \(4 + 11 = \color{brown}\boxed{15}\)

 Jun 26, 2022

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